[boost] more on c2_function discussion

From: Marcus Mendenhall (marcus.h.mendenhall_at_[hidden])
Date: 2006-03-25 21:06:26

>
> Message: 2
> Date: Sat, 25 Mar 2006 12:02:50 -0500
> From: Neal Becker <ndbecker2_at_[hidden]>
> Subject: Re: [boost] c2_functions first submission materials uploaded
> To: boost_at_[hidden]
> Message-ID: <e03t3p$2ok$1_at_[hidden]>
> Content-Type: text/plain; charset=us-ascii
>
> Roland Schwarz wrote:
>
>> Neal Becker wrote:
>>> 1. Generic data types instead of hard-coded 'double'
>> Hmm, generic? int's won't make any sense would they?
>> perhaps double or float but what else?
>>
>
> I was thinking of any float-like type, but that's an interesting  
> question.
> I'm afraid I don't have enough expertise to answer it - do any of  
> these
> algorithms make sense for other numeric fields, (e.g., finite fields)?
>
That's for another package... This one needs to be kept very focused  
on plain floating types.
>>
>> Roland Schwarz wrote:
>>> But complex of course could also turn out to be handy at times.
>> Too fast again (._.)
>> A complex function AFAIK is indefinitely differentiable by  
>> definition.
>>
>> So it obviously belongs to the class of C2 but is there such a
>> thing as
>> complex valued splines?
>>
Actually, if by this one means a complex function of a real  
argument,  the extension is trivial since one can spline the real &  
imaginary parts separately.  Otherwise, it is a bit murky.
> From: Roland Schwarz <roland.schwarz_at_[hidden]>
> Subject: Re: [boost] c2_functions suggestions commentary
> To: boost_at_[hidden]
> Message-ID: <442589C0.8060006_at_[hidden]>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>
> Marcus Mendenhall wrote:
>> For example, for
>> bessel functions, one gets derivatives out of the same recursion
>> relation (when recurring over the order) used to generate the
>> function itself.
> True. But why stop at order 2 ? Wouldn't one rather like to be able to
> evaluate a function to arbitrary order say n? (y, y_1, y_2, ...y_n )
> Then you would supply a
>
> 1) framework of function evaluation that returns a vector of  
> derivatives
> whose size is user specifiable or automatically selected by the  
> algorithm
>
> 2) a set of algorithms that can make use of derivatives of up to  
> order n.
>
> 3) a set of functions that are able to efficiently calculate not only
> the function value but also the required number of derivatives.
>
> E.g.:
>
> sin(Pi/2,5) would return a vector (1,0,-1,0,1)
This is a bang-for-the-buck issue.  In real numerical computations,  
the manipulations get more complex rapidly for higher derivatives,  
but the yield gets less.  Especially since this is aimed  
significantly at physical data, two derivatives is about all one has  
reason to believe in.

I think a general C(N) package is really something for theorists to  
use, but then one can just use Mathematica for that kind of  
calculation any.  Let's not try to rewrite Mathematica with this  
package. I would rather contribute a package which does a focused  
task perfectly and compactly, than a big fuzzy package that no one  
can use because it does too many things halfway.
>

> Message: 6
> Date: Sat, 25 Mar 2006 19:29:10 +0100
> From: Roland Schwarz <roland.schwarz_at_[hidden]>
> Subject: Re: [boost] c2_functions suggestions commentary
> To: boost_at_[hidden]
> Message-ID: <44258BF6.6000203_at_[hidden]>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>
> Marcus Mendenhall wrote:
>> The cubic splines (as opposed to piecewise cubic interpolators) are
>> very smooth at the boundaries, so I think there is not a problem
>> there.  It has worked for me with no difficulties.
>
> Hmm, whether a spline is interpolating or approximating only has to do
> with how you calculated its pivot values. I am not sure if I  
> understand
> you correctly here.
>
> Roland
>
If you look at the source for splint (which derives, a long ways back  
from the code in Numerical Recipes, but is heavily modified), you  
will see the forced continuity at the boundaries, including a way of  
expanding which (I hope) should result in good roundoff behavior at  
the boundary.  Anyway, the root finder uses a numerical tolerance  
which should prevent it from having convergence issues due to the  
last bit anyway.
> ------------------------------
>
> Message: 10
> Date: Sat, 25 Mar 2006 15:03:44 -0500
> From: Jason Hise <chaos_at_[hidden]>
> Subject: Re: [boost] c2_functions first submission materials uploaded
> To: boost_at_[hidden]
> Message-ID: <4425A220.4070900_at_[hidden]>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>>
>>
> Rational numbers?
>
The concept of a continuous function on the rationals is suspect to  
me.  Also, this is focused on numerical computation, not number  
theory.  Everything is a float of some sort, to me!